Tropical and nonarchimedean analytic methods in algebraic geometry

代数几何中的热带和非阿基米德解析方法

基本信息

批准号：
2001502
负责人：
Sam Payne
金额：
$ 35.97万
依托单位：
University of Texas at Austin
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001502&HistoricalAwards=false
关键词：
Tropical nonarchimedean analytic methods algebraic

项目摘要

Algebraic geometry studies solution sets of systems of polynomial equations. For instance, lines are solution sets of linear polynomial equations, while circles and hyperbolas are solution sets to quadratic polynomial equations, and their study goes back to the ancient Greeks. The solution sets of systems of many polynomial equations in many variables often have beautiful and complicated geometry. The PI will apply new and modern techniques to answer questions of classical interest in the field of algebraic geometry, and to address long standing open problems about the geometry of curves defined by polynomial equations. This project provides research training opportunities for graduate students.Over a nonarchimedean field, one can split the problem of understanding such solution sets into two parts. What are the possible valuations of solutions? And what are the solutions with a given valuation? The set of valuations of solutions has a rich combinatorial and polyhedral structure, and is the primary object of study in tropical geometry; the solutions with a given valuation can be studied via nonarchimedean analytic geometry. Recent developments in these fields make it possible to resolve subtle questions about the geometry of the actual solution set using new and innovative tools and techniques. This project will refine, abstract, and generalize these new methods and explore deeper applications to open problems in algebraic geometry.The PI will use tropical and nonarchimedean methods to continue his work on refined curve counting, and on unstable cohomology of moduli spaces of curves, proceeding beyond the top graded piece of the weight filtration to examine the full weight-graded cohomology ring. He will also initiate an analogous study of the weight graded cohomology of the moduli space of abelian varieties, using the topology of moduli spaces of tropical abelian varieties and the combinatorial algebra of complexes built out of unimodular matroids and perfect quadratic forms. Additional projects will attack the motivic, p-adic, and topological monodromy conjectures for Newton nondegenerate singularities, using the combinatorics of relative local h-polynomials and Stapledon’s formulas for monodromy, and attempt to resolve outstanding (and mutually contradictory) conjectures of Kontsevich and Morita in the homology of commutative graph complexes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

多项式方程系统的代数几何研究解决方案集。例如，线是线性多项式方程的解决方案集，而电路和双曲线是跨二次多项式方程的溶液集，它们的研究可以追溯到古希腊人。许多变量中许多多项式方程的系统的解决方案集通常具有美丽而复杂的几何形状。 PI将采用新的现代技术来回答代数几何学领域的经典兴趣问题，并解决有关多项式方程定义的曲线几何形状的长期开放问题。该项目为研究生提供了研究培训机会。在一个非建筑领域，可以将理解此类解决方案集的问题分为两个部分。解决方案的可能值是什么？具有给定价值的解决方案是什么？溶液值集具有丰富的组合和多面体结构，是热带几何学研究的主要对象。具有给定值的溶液可以通过非建筑分析几何形状研究。这些领域的最新发展使得使用新的和创新的工具和技术解决有关实际解决方案集的几何形状的微妙问题。该项目将完善，抽象和概括这些新方法，并探索更深层次的应用程序，以在代数几何形状中进行开放问题。PI将使用热带和非架构的方法继续他的工作在精致的曲线计数上，并在曲线的不稳定的曲线空间上进行曲线的不稳定共识，从而超出了体重的最高体重，以进行体重较高的体重量的完整体重限制。他还将使用热带阿伯利亚品种的莫德利（Modulli）空间的拓扑结构和由单模型的matroids和Perfect Quadratic形式构建的复合物组合代数的拓扑，对阿贝尔品种的莫德利摩德里空间的重量分级共同体学进行了类似的研究。其他项目将使用相对本地的H-多种物质和Stapledon的单构图案的组合来攻击牛顿非排效奇点的图案，p-dadic和拓扑单，并试图解决Kontsevich和Mority in Mortivatient in Mority in the Homelogy in Smotory ns of Homeolation ns of the Homelogy ns of the Homelogy n s of the Homegory n s of Homegies in the Homegory n s of Himoties ns of the Homegory n s of Homegiest的组合。任务，并通过评估诚实地认为，使用基金会的知识分子和更广泛的影响标准。

项目成果

期刊论文数量（6）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Tropical moduli spaces as symmetric Δ$\Delta$‐complexes

作为对称 Î$Delta$âcomplex 的热带模空间

DOI：
10.1112/blms.12570
发表时间：
2022
期刊：
Bulletin of the London Mathematical Society
影响因子：
0.9
作者：
Allcock, Daniel;Corey, Daniel;Payne, Sam
通讯作者：
Payne, Sam

Compactified Jacobians as Mumford models

压缩雅可比行列式作为 Mumford 模型

DOI：
10.1090/tran/8875
发表时间：
2023
期刊：
Transactions of the American Mathematical Society
影响因子：
1.3
作者：
Christ, Karl;Payne, Sam;Shen, Tif
通讯作者：
Shen, Tif

Equivariant Grothendieck–Riemann–Roch andlocalization in operational K-theory

等变格洛腾迪克-黎曼-罗赫和可操作 K 理论中的局域化

DOI：
10.2140/ant.2021.15.341
发表时间：
2021
期刊：
Algebra & Number Theory
影响因子：
1.3
作者：
Anderson, Dave;Gonzales, Richard;Payne, Sam
通讯作者：
Payne, Sam

Bitangents to plane quartics via tropical geometry: rationality, $$\mathbb {A}^1$$-enumeration, and real signed count

通过热带几何到平面四次曲线的双切线：理性、$$mathbb {A}^1$$-枚举和实数有符号数

DOI：
10.1007/s40687-023-00383-1
发表时间：
2023
期刊：
Research in the Mathematical Sciences
影响因子：
1.2
作者：
Markwig, Hannah;Payne, Sam;Shaw, Kris
通讯作者：
Shaw, Kris

Triangulations of simplices with vanishing local h-polynomial

具有消失局部 h 多项式的单纯形三角剖分

DOI：
10.5802/alco
发表时间：
2020
期刊：
Algebraic combinatorics
影响因子：
0
作者：
de Moura, André;Gunther, Elijah;Payne, Sam;Schuchardt, Jason;Stapledon, Alan
通讯作者：
Stapledon, Alan